On the subgroup structure of the hyperoctahedral group in six dimensions.

The subgroup structure of the hyperoctahedral group in six dimensions is investigated. In particular, the subgroups isomorphic to the icosahedral group are studied. The orthogonal crystallographic representations of the icosahedral group are classified and their intersections and subgroups analysed, using results from graph theory and their spectra.

Introduction

The discovery of quasicrystals in 1984 ▶ by Shechtman et al. has spurred the mathematical and physical community to develop mathematical tools in order to study structures with non­crystallographic symmetry.

Quasicrystals are alloys with five-, eight-, ten- and 12-fold symmetry in their atomic positions (Steurer, 2004 ▶), and therefore they cannot be organized as (periodic) lattices. In crystallographic terms, their symmetry group G is noncrystallographic. However, the noncrystallographic symmetry leaves a lattice invariant in higher dimensions, providing an integral representation of G. If such a representation is reducible and contains a two- or three-dimensional invariant subspace, then it is referred to as a crystallographic representation, following terminology given by Levitov & Rhyner (1988 ▶). This is the starting point to construct quasicrystals via the cut-and-project method described by, among others, Senechal (1995 ▶), or as a model set (Moody, 2000 ▶).

In this paper we are interested in icosahedral symmetry. The icosahedral group consists of all the rotations that leave a regular icosahedron invariant, it has size 60 and it is the largest of the finite subgroups of . contains elements of order five, therefore it is noncrystallographic in three dimensions; the (minimal) crystallographic representation of it is six-dimensional (Levitov & Rhyner, 1988 ▶). The full icosahedral group, denoted by , also contains the reflections and is equal to , where denotes the cyclic group of order two. is isomorphic to the Coxeter group (Humphreys, 1990 ▶) and is made up of 120 elements. In this work, we focus on the icosahedral group I because it plays a central role in applications in virology (Indelicato et al., 2011 ▶). However, our considerations apply equally to the larger group .

Levitov & Rhyner (1988 ▶) classified the Bravais lattices in that are left invariant by : there are, up to equivalence, exactly three lattices, usually referred to as icosahedral Bravais lattices, namely the simple cubic (SC), body-centred cubic (BCC) and face-centred cubic (FCC). The point group of these lattices is the six-dimensional hyperoctahedral group, denoted by , which is a subgroup of and can be represented in the standard basis of as the set of all orthogonal and integral matrices. The subgroups of which are isomorphic to the icosahedral group constitute the integral representations of it; among them, the crystallographic ones are those which split, in , into two three-dimensional irreducible representations of . Therefore, they carry two subspaces in which are invariant under the action of and can be used to model the quasiperiodic structures.

The embedding of the icosahedral group into has been used extensively in the crystallographic literature. Katz (1989 ▶), Senechal (1995 ▶), Kramer & Zeidler (1989 ▶), Baake & Grimm (2013 ▶), among others, start from a six-dimensional crystallographic representation of to construct three-dimensional Penrose tilings and icosahedral quasicrystals. Kramer (1987 ▶) and Indelicato et al. (2011 ▶) also apply it to study structural transitions in quasicrystals. In particular, Kramer considers in a representation of and a representation of the octahedral group which share a tetrahedral subgroup, and defines a continuous rotation (called Schur rotation) between cubic and icosahedral symmetry which preserves intermediate tetrahedral symmetry. Indelicato et al. define a transition between two icosahedral lattices as a continuous path connecting the two lattice bases keeping some symmetry preserved, described by a maximal subgroup of the icosahedral group. The rationale behind this approach is that the two corresponding lattice groups share a common subgroup. These two approaches are shown to be related (Indelicato et al., 2012 ▶), hence the idea is that it is possible to study the transitions between icosahedral quasicrystals by considering two distinct crystallographic representations of in which share a common subgroup.

These papers motivate the idea of studying in some detail the subgroup structure of . In particular, we focus on the subgroups isomorphic to the icosahedral group and its subgroups. Since the group is quite large (it has elements), we use for computations the software GAP (The GAP Group, 2013 ▶), which is designed to compute properties of finite groups. More precisely, based on Baake (1984 ▶), we generate the elements of in GAP as a subgroup of the symmetric group and then find the classes of subgroups isomorphic to the icosahedral group. Among them we isolate, using results from character theory, the class of crystallographic representations of . In order to study the subgroup structure of this class, we propose a method using graph theory and their spectra. In particular, we treat the class of crystallographic representations of as a graph: we fix a subgroup of and say that two elements in the class are adjacent if their intersection is equal to a subgroup isomorphic to . We call the resulting graph -graph. These graphs are quite large and difficult to visualize; however, by analysing their spectra (Cvetkovic et al., 1995 ▶) we can study in some detail their topology, hence describing the intersection and the subgroups shared by different representations.

The paper is organized as follows. After recalling, in §2, the definitions of point group and lattice group, we define, in §3, the crystallographic representations of the icosahedral group and the icosahedral lattices in six dimensions. We provide, following Kramer & Haase (1989 ▶), a method for the construction of the projection into three dimensions using tools from the representation theory of finite groups. In §4 we classify, with the help of GAP, the crystallographic representations of . In §5 we study their subgroup structure, introducing the concept of -graph, where is a subgroup of .

Lattices and noncrystallographic groups

Let , be a basis of , and let be the matrix whose columns are the components of with respect to the canonical basis of . A lattice in is a -free module of rank n with basis B, i.e. Any other lattice basis is given by BM, where , the set of invertible matrices with integral entries (whose determinant is equal to ) (Artin, 1991 ▶).

The point group of a lattice is given by all the orthogonal transformations that leave the lattice invariant (Pitteri & Zanzotto, 2002 ▶):

We notice that, if , then . In other words, the point group consists of all the orthogonal matrices which can be represented in the basis B as integral matrices. The set of all these matrices constitute the lattice group of the lattice:

The lattice group provides an integral representation of the point group and these are related via the equation and moreover the following hold (Pitteri & Zanzotto, 2002 ▶):

We notice that a change of basis in the lattice leaves the point group invariant, whereas the corresponding lattice groups are conjugated in . Two lattices are inequivalent if the corresponding lattice groups are not conjugated in (Pitteri & Zanzotto, 2002 ▶).

As a consequence of the crystallographic restriction [see, for example, Baake & Grimm (2013 ▶)] five- and n-fold symmetries, where n is a natural number greater than six, are forbidden in dimensions two and three, and therefore any group G containing elements of such orders cannot be the point group of a two- or three-dimensional lattice. We therefore call these groups noncrystallographic. In particular, three-dimensional icosahedral lattices cannot exist. However, a noncrystallographic group leaves some lattices invariant in higher dimensions and the smallest such dimension is called the minimal embedding dimension. Following Levitov & Rhyner (1988 ▶), we introduce:

Definition 2.1

Let G be a noncrystallographic group. A crystallographic representation ρ of G is a D-dimensional representation of G such that:

(1) the characters of ρ are integers;

(2) ρ is reducible and contains a two- or three-dimensional representation of G.

We observe that the first condition implies that G must be the subgroup of the point group of a D-dimensional lattice. The second condition tells us that ρ contains either a two- or three-dimensional invariant subspace E of , usually referred to as physical space (Levitov & Rhyner, 1988 ▶).

Six-dimensional icosahedral lattices

The icosahedral group is generated by two elements, and , such that , where e denotes the identity element. It has size 60 and it is isomorphic to , the alternating group of order five (Artin, 1991 ▶). Its character table is given in Table 1 ▶.

Table 1

Character table of the icosahedral group

Note that is the golden ratio.

From the character table we see that the (minimal) crystallographic representation of is six-dimensional and is given by . Therefore, leaves a lattice in invariant. Levitov & Rhyner (1988 ▶) proved that the three inequivalent Bravais lattices of this type, mentioned in the Introduction and referred to as icosahedral (Bravais) lattices, are given by, respectively:

We note that a basis of the SC lattice is the canonical basis of . Its point group is given by which is the hyperoctahedral group in dimension six. In the following, we will denote this group by , following Humphreys (1996 ▶). We point out that this notation comes from Lie theory: indeed, represents the root system of the Lie algebra (Fulton & Harris, 1991 ▶). However, the corresponding reflection group is isomorphic to the hyperoctahedral group in six dimensions (Humphreys, 1990 ▶).

All three lattices have point group , whereas their lattice groups are different and, indeed, they are not conjugate in (Levitov & Rhyner, 1988 ▶).

Let be a subgroup of isomorphic to . provides a (faithful) integral and orthogonal representation of . Moreover, if in , then is also crystallographic (in the sense of Definition 2.1). All of the other crystallographic representations are given by , where is a basis of an icosahedral lattice in . Therefore we can focus our attention, without loss of generality, on the orthogonal crystallographic representations.

Projection operators

Let be a crystallographic representation of the icosahedral group. splits into two three-dimensional irreducible representations (IRs), and , in . This means that there exists a matrix such that

The two IRs and leave two three-dimensional subspaces invariant, which are usually referred to as the physical (or parallel) space and the orthogonal space (Katz, 1989 ▶). In order to find the matrix R (which is not unique in general), we follow (Kramer & Haase, 1989 ▶) and use results from the representation theory of finite groups (for proofs and further results see, for example, Fulton & Harris, 1991 ▶). In particular, let be an n-dimensional representation of a finite group G over a field F (F = , ). By Maschke’s theorem, Γ splits, in , as , where is an -dimensional IR of G. Then the projection operator is given by where denotes the complex conjugate of the character of the representation . This operator is such that its image is equal to an -dimensional subspace of invariant under . In our case, we have two projection operators, , i = 1, 2, corresponding to the IRs and , respectively. We assume the image of , , to be equal to , and . If is the canonical basis of , then a basis of (respectively ) can be found considering the set for (respectively ) and then extracting a basis from it. Since = , we obtain , for i = 1, 2. The matrix R can be thus written as

Denoting by and the matrices which represent and in the bases and , respectively, we have, by linear algebra

Since [cf. equation (2)], we obtain for all and . In particular, the following diagram commutes

The set is the starting point for the construction of quasicrystals via the cut-and-project method (Senechal, 1995 ▶; Indelicato et al., 2012 ▶).

Crystallographic representations of

From the previous section it follows that the six-dimensional hyperoctahedral group contains all the (minimal) orthogonal crystallographic representations of the icosahedral group. In this section we classify them, with the help of the computer software programme GAP (The GAP Group, 2013 ▶).

Representations of the hyperoctahedral group

Permutation representations of the n-dimensional hyperoctahedral group in terms of elements of , the symmetric group of order 2n, have been described by Baake (1984 ▶). In this subsection, we review these results because they allow us to generate in GAP and further study its subgroup structure.

It follows from equation (1) that consists of all the orthogonal integral matrices. A matrix of this kind must satisfy , the identity matrix of order six, and have integral entries only. It is easy to see that these conditions imply that A has entries in {0, ±1} and each row and column contains 1 or −1 only once. These matrices are called signed permutation matrices. It is straightforward to see that any can be written in the form NQ, where Q is a permutation matrix and N is a diagonal matrix with each diagonal entry being either 1 or −1. We can thus associate with each matrix in a pair , where is a vector given by the diagonal elements of N, and is the permutation associated with Q. The set of all these pairs constitutes a group (called the wreath product of and , and denoted by ; Humphreys, 1996 ▶) with the multiplication rule given by where denotes addition modulo 2 and and are isomorphic, an isomorphism T being the following:

It immediately follows that |B 6| = 266! = 46 080. A set of generators is given by which satisfy the relations

Finally, the function defined by is injective and maps any element of into a permutation of , and provides a faithful permutation representation of as a subgroup of . Combining equation (8) with the inverse of equation (10) we get the function which can be used to map a permutation into an element of .

Classification

In this subsection we classify the orthogonal crystallographic representations of the icosahedral group. We start by recalling a standard way to construct such a representation, following Zappa et al. (2013 ▶). We consider a regular icosahedron and we label each vertex by a number from one to 12, so that the vertex opposite to vertex i is labelled by (see Fig. 1 ▶). This labelling induces a permutation representation given by Using equation (11) we obtain a representation given by

Figure 1

A planar representation of an icosahedral surface, showing our labelling convention for the vertices; the dots represent the locations of the symmetry axes corresponding to the generators of the icosahedral group and its subgroups. The kite highlighted is a fundamental domain of the icosahedral group.

We see that and , so that, by looking at the character table of , we have which implies, using Maschke’s theorem (Fulton & Harris, 1991 ▶), that in . Therefore, the subgroup of is a crystallographic representation of .

Before we continue, we recall the following (Humphreys, 1996 ▶):

Definition 4.1

Let H be a subgroup of a group G. The conjugacy class of H in G is the set

In order to find all the other crystallographic representations, we use the following scheme:

(a) we generate as a subgroup of using equations (9) and (10);

(b) we list all the conjugacy classes of the subgroups of and find a representative for each class;

(c) we isolate the classes whose representatives have order 60;

(d) we check if these representatives are isomorphic to ;

(e) we map these subgroups of into using equation (11) and isolate the crystallographic ones by checking the characters; denoting by S the representative, we decompose as

Note that S is crystallographic if and only if and .

We implemented steps (1)–(4) in GAP (see Appendix C). There are three conjugacy classes of subgroups isomorphic to in . Denoting by the representatives of the classes returned by GAP, we have, using equation (11),

Since 2A is decomposable into two one-dimensional representations, it is not strictly speaking two dimensional in the sense of Definition 2.1, and as a consequence, only the second class contains the crystallographic representations of . A computation in GAP shows that its size is 192. We thus have the following:

Proposition 4.1

The crystallographic representations of in form a unique conjugacy class in the set of all the classes of subgroups of , and its size is equal to 192.

We briefly point out that the other two classes of subgroups isomorphic to in have an interesting algebraic intepretation. First of all, we observe that is an extension of , since according to Humphreys (1996 ▶):

Following Janusz & Rotman (1982 ▶), it is possible to embed the symmetric group into in two different ways. The canonical embedding is achieved by fixing a point in and permuting the other five, whereas the other embedding is by means of the so-called ‘exotic map’ , which acts on the six 5-Sylow subgroups of by conjugation. Recalling that the icosahedral group is isomorphic to the alternating group , which is a normal subgroup of , then the canonical embedding corresponds to the representation in , while the exotic one corresponds to the representation .

In what follows, we will consider the subgroup previously defined as a representative of the class of the crystallographic representations of , and denote this class by .

Recalling that two representations and of a group G are said to be equivalent if there are related via a similarity transformation, i.e. there exists an invertible matrix S such that then an immediate consequence of Proposition 4.1 is the following:

Corollary 4.1

Let and be two orthogonal crystallographic representations of . Then and are equivalent in .

We observe that the determinant of the generators of in equation (12) is equal to one, so that . Proposition 4.1 implies that all the crystallographic representations belong to . The remarkable fact is that they split into two different classes in . To see this, we first need to generate . In particular, with GAP we isolate the subgroups of index two in , which are normal in , and then, using equation (11), we find the one whose generators have determinant equal to one. In particular, we have

We can then apply the same procedure to find the crystallographic representations of , and see that they split into two classes, each one of size 96. Again we can choose as a representative for one of these classes; a representative for the other one is given by

We note that in the more general case of , we can construct the crystallographic representations of starting from the crystallographic representations of . First of all, we recall that , where is the cyclic group of order two. Let be a crystallographic representation of in , and let be a one-dimensional representation of . Then the representation given by where denotes the tensor product of matrices, is a representation of in and it is crystallographic in the sense of Definition 2.1 (Fulton & Harris, 1991 ▶).

Projection into the three-dimensional space

We study in detail the projection into the physical space using the methods described in §3.1.

Let be the crystallographic representation of given in equation (12). Using equation (3) with and we obtain the following projection operators The rank of these operators is equal to three. We choose as a basis of and the following linear combination of the columns of the projection operators , for i = 1, 2 and :

With a suitable rescaling, we obtain the matrix R given by

The matrix R is orthogonal and reduces as in equation (2). In Table 2 ▶ we give the explicit forms of the reduced representation. The matrix representation in of is given by [see equation (5)]

Table 2

Explicit forms of the IRs and with

Generator	Irrep	Irrep

The orbit , where is the canonical basis of , represents a regular icosahedron in three dimensions centred at the origin (Senechal, 1995 ▶; Katz, 1989 ▶; Indelicato et al., 2011 ▶).

Let be another crystallographic representation of in . By Proposition 4.1, and are conjugated in . Consider such that and let . We have Therefore it is possible, with a suitable choice of the reducing matrices, to project all the crystallographic representations of in in the same physical space.

Subgroup structure

The nontrivial subgroups of are listed in Table 3 ▶, together with their generators (Hoyle, 2004 ▶). Note that , and are maximal subgroups of , and that , and are normal subgroups of , and , respectively (Humphreys, 1996 ▶; Artin, 1991 ▶). The permutation representations of the generators in are given in Table 4 ▶ (see also Fig. 1 ▶).

Table 3

Nontrivial subgroups of the icosahedral group

stands for the tetrahedral group, for the dihedral group of size 2n, and C for the cyclic group of size n.

Subgroup	Generators	Relations	Size
			12
			10
			6
			5
			4
			3
			2

Table 4

Permutation representations of the generators of the subgroups of the icosahedral group

Since is a small group, its subgroup structure can be easily obtained in GAP by computing explicitly all its conjugacy classes of subgroups. In particular, there are seven classes of nontrivial subgroups in : any subgroup H of has the property that, if K is another subgroup of isomorphic to H, then H and K are conjugate in (this property is referred to as the ‘friendliness’ of the subgroup H; Soicher, 2006 ▶). In other words, denoting by the number of subgroups of isomorphic to , i.e. we have (cf. Definition 4.1)

In Table 5 ▶ we list the size of each class of subgroups in . Geometrically, different copies of , and correspond to the different two-, three- and fivefold axes of the icosahedron, respectively. In particular, different copies of stabilize one of the six fivefold axes of the icosahedron, and each copy of stabilizes one of the ten threefold axes. Moreover, it is possible to inscribe five tetrahedra into a dodecahedron, and each different copy of the tetrahedral group in stabilizes one of these tetrahedra.

Table 5

Sizes of the classes of subgroups of the icosahedral group in and B 6

Subgroup
	5	480
	6	576
	10	960
	5	120
	6	576
	10	320
	15	180

Subgroups of the crystallographic representations of

Let be a subgroup of . The function (11) provides a representation of in , denoted by , which is a subgroup of . Let us denote by the conjugacy class of in . The next lemma shows that this class contains all the subgroups of the crystallographic representations of in .

Lemma 5.1

Let be a crystallographic representation of in and let be a subgroup of isomorphic to . Then .

Proof

Since , there exists such that , and therefore is a subgroup of isomorphic to . Since all these subgroups are conjugate in [they are ‘friendly’ in the sense intended by Soicher (2006 ▶)], there exists such that . Thus , implying that .

□

We next show that every element of is a subgroup of a crystallographic representation of .

Lemma 5.2

Let . There exists such that is a subgroup of .

Since , there exists such that . We define . It can be seen immediately that is a subgroup of .

□

As a consequence of these lemmata, contains all the subgroups of which are isomorphic to and are subgroups of a crystallographic representation of . The explicit forms of are given in Appendix B. We point out that it is possible to find subgroups of isomorphic to a subgroup of which are not subgroups of any crystallographic representation of . For example, the following subgroup is isomorphic to the tetrahedral group ; a computation in GAP shows that it is not a subgroup of any elements in . Indeed, the two classes of subgroups, and , are disjoint.

Using GAP, we compute the size of each (see Table 5 ▶). We observe that . This implies that crystallographic representations of may share subgroups. In order to describe more precisely the subgroup structure of we will use some basic results from graph theory and their spectra, which we are going to recall in the next section.

Some basic results of graph theory and their spectra

In this section we recall, without proofs, some concepts and results from graph theory and spectral graph theory. Proofs and further results can be found, for example, in Foulds (1992 ▶) and Cvetkovic et al. (1995 ▶).

Let G be a graph with vertex set . The number of edges incident with a vertex v is called the degree of v. If all vertices have the same degree d, then the graph is called regular of degree d. A walk of length l is a sequence of l consecutive edges and it is called a path if they are all distinct. A circuit is a path starting and ending at the same vertex and the girth of the graph is the length of the shortest circuit. Two vertices p and q are connected if there exists a path containing p and q. The connected component of a vertex v is the set of all vertices connected to v.

The adjacency matrix A of G is the matrix whose entries are equal to one if the vertex is adjacent to the vertex , and zero otherwise. It can be seen immediately from its definition that A is symmetric and for all i, so that . It follows that A is diagonalisable and all its eigenvalues are real. The spectrum of the graph is the set of all the eigenvalues of its adjacency matrix A, usually denoted by .

Theorem 5.1

Let A be the adjacency matrix of a graph G with vertex set . Let denote the number of walks of length k starting at vertex and finishing at vertex . We have

We recall that the spectral radius of a matrix A is defined by . If A is a non-negative matrix, i.e. if all its entries are non-negative, then (Horn & Johnson, 1985 ▶). Since the adjacency matrix of a graph is non-negative, , where and r is the largest eigenvalue. r is called the index of the graph G.

Theorem 5.2

Let be the spectrum of a graph G, and let r denote its index. Then G is regular of degree r if and only if Moreover, if G is regular the multiplicity of its index is equal to the number of its connected components.

Applications to the subgroup structure

Let be a subgroup of . In the following we represent the subgroup structure of the class of crystallographic representations of in , , as a graph. We say that are adjacent to each other (i.e. connected by an edge) in the graph if there exists such that . We can therefore consider the graph , where an edge is of the form . We call this graph -graph.

Using GAP, we compute the adjacency matrices of the -graphs. The algorithms used are shown in Appendix C. The spectra of the -graphs are given in Table 6 ▶. We first of all notice that the adjacency matrix of the -graph is the null matrix, implying that there are no two representations sharing precisely a subgroup isomorphic to , i.e. not a subgroup containing . We point out that, since the adjacency matrix of the -graph is not the null one, then there exist crystallographic representations, say and , sharing a maximal subgroup isomorphic to . Since is a (normal) subgroup of , then and do share a subgroup, but also a subgroup. In other words, if two representations share a fivefold axis, then necessarily they also share a twofold axis.

A straightforward calculation based on Theorem 5.2 leads to the following

Proposition 5.1

Let be a subgroup of . Then the corresponding -graph is regular.

In particular, the degree of each -graph is equal to the largest eigenvalue of the corresponding spectrum. As a consequence we have the following:

Proposition 5.2

Let be a crystallographic representation of in . Then there are exactly representations such that In particular, we have = 5, 6, 10, 0, 30, 20, 60 and 60 for , and , respectively.

In particular, this means that for any crystallographic representation of there are precisely other such representations which share a subgroup isomorphic to . In other words, we can associate to the class the ‘subgroup matrix’ S whose entries are defined by

The matrix S is symmetric and , for all i, since the order of is 60. It follows from Proposition 5.2 that each row of S contains entries equal to . Moreover, a rearrangement of the columns of S shows that the 192 crystallographic representations of can be grouped into 12 sets of 16 such that any two of these representations in such a set of 16 share a -subgroup. This implies that the corresponding subgraph of the -graph is a complete graph, i.e. every two distinct vertices are connected by an edge. From a geometric point of view, these 16 representations correspond to ‘six-dimensional icosahedra’. This ensemble of 16 such icosahedra embedded into a six-dimensional hypercube can be viewed as a six-dimensional analogue of the three-dimensional ensemble of five tetrahedra inscribed into a dodecahedron, sharing pairwise a -subgroup.

We notice that, using Theorem 5.2, not all the graphs are connected. In particular, the - and the -graphs are made up of six connected components, whereas the - and the -graphs consist of two connected components. With GAP, we implemented a breadth-first search algorithm (Foulds, 1992 ▶), which starts from a vertex i and then ‘scans’ for all the vertices connected to it, which allows us to find the connected components of a given -graph (see Appendix C). We find that each connected component of the - and -graphs is made up of 32 vertices, while for the - and -graphs each component consists of 96 vertices. For all other subgroups, the corresponding -graph is connected and the connected component contains trivially 192 vertices.

We now consider in more detail the case when is a maximal subgroup of . Let and let us consider its vertex star in the corresponding -graph, i.e.

A comparison of Tables 5 ▶ and 6 ▶ shows that [i.e. the number of subgroups isomorphic to in , cf. equation (14)] and therefore, since the graph is regular, . This suggests that there is a one-to-one correspondence between elements of the vertex star of and subgroups of isomorphic to ; in other words, if we fix any subgroup P of isomorphic to , then P ‘connects’ with exactly another representation . We thus have the following:

Proposition 5.3

Let be a maximal subgroup of . Then for every there exist exactly two crystallographic representations of , , such that .

In order to prove it, we first need the following lemma:

Lemma 5.3

Let be a maximal subgroup of . Then the corresponding -graph is triangle-free, i.e. it has no circuits of length three.

Let be the adjacency matrix of the -graph. By Theorem 5.1, its third power determines the number of walks of length three, and in particular its diagonal entries, , for , correspond to the number of triangular circuits starting and ending in vertex i. A direct computation shows that , for all i, thus implying the non-existence of triangular circuits in the graph.

□

Proof of Proposition 5.3

If , then, using Lemma 5.2, there exists such that P is a subgroup of . Let us consider the vertex star . We have ; we call its elements . Let us suppose that P is not a subgroup of any , for . This implies that P does not connect with any of these . However, since has exactly different subgroups isomorphic to , then at least two vertices in the vertex star, say and , are connected by the same subgroup isomorphic to , which we denote by Q. Therefore we haveThis implies that , and form a triangular circuit in the graph, which is a contradiction due to Lemma 5.3, hence the result is proved.

□

It is noteworthy that the situation in is different. If we denote by and the two disjoint classes of crystallographic representations of in [cf. equation (13)], we can build, in the same way as described before, the -graphs for and , for and . The result is that the adjacency matrices of all these six graphs are the null matrix of dimension 96. This implies that these graphs have no edges, and so the representations in each class do not share any maximal subgroup of . As a consequence, we have the following:

Proposition 5.4

Let be two crystallographic representations of , and , , where is a maximal subgroup of . Then and are not conjugated in . In other words, the elements of which conjugate with are matrices with determinant equal to −1.

We conclude by showing a computational method which combines the result of Propositions 4.1 and 5.2. We first recall the following:

Definition 5.1

Let H be a subgroup of a group G. The normaliser of H in G is given by

Corollary 5.1

Let and be two crystallographic representations of in and such that . Let be the set of all the elements of which conjugate with and let be the normaliser of P in . We have In other words, it is possible to find a nontrivial element in the normaliser of P in which conjugates with .

Let us suppose . Then , for all . This implies, since , that P is not a subgroup of , which is a contradiction.

□

We give now an explicit example. We consider the representation as in equation (12), and its subgroup (the explicit form is given in Appendix B). With GAP, we find the other representation such that . Its explicit form is given by

A direct computation shows that the matrix belongs to and conjugate with . Note that .

Conclusions

In this work we explored the subgroup structure of the hyperoctahedral group in six dimensions. In particular we found the class of the crystallographic representations of the icosahedral group, whose size is 192. Any such representation, together with its corresponding projection operator , can be chosen to construct icosahedral quasicrystals via the cut-and-project method. We then studied in detail the subgroup structure of this class. For this, we proposed a method based on spectral graph theory and introduced the concept of -graph, for a subgroup of the icosahedral group. This allowed us to study the intersection and the subgroups shared by different representations. We have shown that, if we fix any representation in the class and a maximal subgroup P of , then there exists exactly one other representation in the class such that . As explained in the Introduction , this can be used to describe transitions which keep intermediate symmetry encoded by P. In particular, this result implies in this context that a transition from a structure arising from via projection will result in a structure obtainable for via projection if the transition has intermediate symmetry described by P. Therefore, this setting is the starting point to analyse structural transitions between icosahedral quasicrystals, following the methods proposed in Kramer (1987 ▶), Katz (1989 ▶) and Indelicato et al. (2012 ▶), which we are planning to address in a forthcoming publication.

These mathematical tools also have many applications in other areas. A prominent example is virology. Viruses package their genomic material into protein containers with regular structures that can be modelled via lattices and group theory. Structural transitions of these containers, which involve rearrangements of the protein lattices, are important in rendering certain classes of viruses infective. As shown in Indelicato et al. (2011 ▶), such structural transitions can be modelled using projections of six-dimensional icosahedral lattices and their symmetry properties. The results derived here therefore have a direct application to this scenario, and the information on the subgroup structure of the class of crystallographic representations of the icosahedral group and their intersections provides information on the symmetries of the capsid during the transition.

The numbers highlighted are the indices of the graphs, and correspond to their degrees .

-graph		-graph		-graph		-graph
Eig.	Mult.	Eig.	Mult.	Eig.	Mult.	Eig.	Mult.
5	1	6	6	10	6	0	192
3	45	2	90	2	90
−3	45	−2	90	−2	90
1	50	−6	6	−10	6
−1	50
−5	1

-graph		-graph		-graph		-graph
Eig.	Mult.	Eig.	Mult.	Eig.	Mult.	Eig.	Mult.
30	1	20	2	60	2	60	1
18	5	4	90	4	90	12	5
12	5	−4	100	−4	90	4	90
6	15			−12	10	−4	90
2	45					−12	5
	31					−60	1
−2	30
−4	45
−8	15